Line integral of a vector field.

Hi all, i'm new to the forums so if i do something stupid don't hesitate to tell me.

Anyway i'm struggling with this problem:

I could do part a ok, but part b has me stumped, I am in the second year of a physics degree and this is a from a maths problem sheet, i haven't done line integrals before now and they have me a bit confused, my textbook has a few examples but none of them include vectors and wiki has me even more confused.

Here is my attempt so far:

(please excuse bad handwriting, i am dyslexic)

I basically don't know where to begin with it, any help much appreciated.

The answer is right. Since dphi/dx=2x-y^2, that means phi=x^2-x*y^2+f(y). Yeah, that is 'integral dx' with f(y) being the constant of integration. That gives dphi/dy=-2xy+f'(y). Since you are supposed to get 6y^2-2xy, you can figure that the f'(y) part must be the 6y^2, so if you put it all together, phi=x^2-xy^2+2y^3.

Well i've managed to do the problem, and i could probably do others now using that technique, but i'm not sure i understand why it works, i mean in questions b, c and d the only constant is x1=0, y1=0, x2=1 and y2=1, why doesn't the chosen path have an effect, is it something to do with the curl being 0?

It has everything to do with the curl being zero. It's one of the conditions you need for V to be conservative. 'Conservative' means that the line integral between two points is independent of the path chosen. If they hadn't given you a conservative V, then the answer would depend on the path.